You know what time it is, it's time for controversial opinions.
In my personal experience, "greek" style seems to work best with a short (50.8 cm/20 in) and less flexible sling. However, I don't like using it. I can get the same or better results with that sling with some other style.
Underarm is more usable, but it is personally an awkward arm motion. One thing I've heard people say before that underarm is for more range, but I highly doubt that. I can get higher speeds and can sling at different angles with other styles. It's neat, but in my experience it doesn't have much purpose.
Anyways, this related to a topic that I've wanted to talk about, but I've already recently made a couple of topics recently so I'm holding off. I'll summarize my thoughts.
Again these are my personal (Emphasis on this part!) perceptions and speculations.
For styles like "Greek" style and "Apache", I don't think they're "real styles". What I mean by "real styles" is that I don't think I they were ever used historically. While I don't know the origins for Apache, I have looked at a number of ancient depictions from Etruscan tombs to the near east (Kapara slinger relief) that depict the "Greek style". The reason I have this in quotations is not because it isn't Greek, but that I don't think these depictions are showing what we call "Greek style".
Let me explain. First of all, is doing it myself. Although I don't have the experience of others, I have been slinging for a while. In that time, I've tried out the Greek style (overhand and sidearm). My impression is... not good. The problem I run into is speed/power. In my experience, starting at the front is a suboptimal position. Although it's the main throw that gives it power, I feel it's harder to get to that point. The other potential indicator is in the art. In most examples I've seen, the sling is actually not above the head (as many do Greek style today), but to the side. Here are a couple examples:
https://artsandculture.google.com/asset/stone-relief-of-a-slinger-from-the-palace-of-king-kapara/7wE6wY7SpMgSMQ?hl=en
(Copy and paste the link or look up "King Kapara slinging relief")
https://coins.www.collectors-society.com/wcm/CoinView.aspx?sc=319922http://slinging.org/forum/YaBB.pl?num=1577297750/9They can be positioned to either side of the head, although the ones with the sling on the opposite side from the throwing arm are somewhat awkward to me.
So what does this depict? My speculation is that it is both a pose and a starting point for one or more additional spins.
Remember how I said that the opposite side of the head from the throwing arm was a bit awkward? Well, it sort of has a parallel. The Polynesian style has the sling across the back. I can't remember where I read this (maybe it was on this website), but I read that the behind the back starting position leads into a single rotation, which then leads into the shot. I've actually tried this, and it works well. For background, my preferred style is the single rotation (Byzantine), so this wasn't much of an adjustment. I'm not saying the original "Greek style" was like this, as it could have been used with any amount of rotations.
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Is Greek a bad style? I'd say yes, but it's kind of neat that this exists. It's also something that probably isn't used to its full potential, but that's probably because relatively few people probably invest into this technique. It might be for good reason that people don't do it, but it might be nice to try something different.
My position on Apache and Underarm are quite similar. They're cool and it's always great to try something new, but there is probably a reason they go unloved. I personally find them to have drawbacks in the way they work with the body.
However, if someone did get good with any of these or some other unconventional style I would love to see it.